# Application

Typical problems addressed are scheduling production, managing complex distribution systems, choosing locations for production facilities, and maintaining the quality of physical goods and services. Other applications includes

• Scheduling – of resources, of vehicles in supply chains and of orders in a factory.
• Facility Planning – Simulations of facility for the rapid and safe processing in factories.
• Planning and Forecasting – identifying possible future developments and deciding how much capacity is needed in future.
• Yield management – Setting the prices of products or services to reflect changing demand and risk.
• Marketing – evaluating the value of sale promotions, developing customer profiles and computing the life-time value of a customer.

One important OR technique is optimization. Many problems in industry require finding the maximum or minimum of an objective function of a set of decision variables, subject to a set of constraints on those variables. Typical objectives are maximum profit, minimum cost, or minimum delay. Frequently there are many decision variables and the solution is not obvious. Techniques of mathematical programming for optimization include linear programming (optimization where both the objective function and constraints depend linearly on the decision variables), non-linear programming (non-linear objective function or constraints), integer programming (decision variables restricted to integer solutions), stochastic programming (uncertainty in model parameter values) and dynamic programming (stage-wise, nested, and periodic decision-making).

Another OR application is the performance analysis of a production line. A typical production line consists of a series of workstations that perform different operations. Jobs flow through the line to be processed at each station. Buffers between stations hold the output of one station and allow it to wait as input to the next. A finite buffer can fill and block output from an upstream station or can empty and starve a downstream station for input. Blocking and starving are key mechanisms of the complex interactions between queues that form in the line. A critical measure of performance is throughput, defined as number of jobs per unit time that can flow through the line. Throughput is reduced when stations experience random machine failures, a common practical situation. Mathematical modeling is needed to capture the impact on throughput of station reliabilities, as well as processing rates and buffer sizes. A model can support operating decisions, such as how to improve a line to meet a throughput target, how to identify bottlenecks, and how much buffer space to allocate in line design.

Another real-world mathematical problem, common to many industries, is the distribution of material and products from plants to customers. For a network of origins and destinations, there are many shipping alternatives, including choices of transportation mode (e.g., road, rail, air) and geographical routes. Some key decisions are routing options over the network, and shipping frequencies on network links. Routing options involve shipping direct, via a terminal or distribution center, and by a combination of routes. These options affect distances traveled and times in transit, which in turn affect transportation and inventory costs. Shipping frequency decisions also affect these costs. Transportation costs favor large infrequent shipments, while inventory costs favor small frequent shipments. Trade-offs between these costs are complex for large networks, and finding the optimal solution is a challenging mathematical problem. In addition to decisions for operations of a given network, there are major strategic decisions, such as the selection and location of distribution centers.

Other OR topics requiring mathematical analysis are inventory control (when to reorder material to avoid shortages under demand uncertainty), manufacturing operations (what size of production run will minimize sum of inventory and production setup costs), location planning (where to locate the hub to serve markets with minimal travel distances), and facility layout (how to design airport terminals to minimize walking distances, maximize number of gates, allow for future expansion, and conform to government regulations).